<aside> 🧭

SOL HANDOFF · COMPLETED TRANSIT

The Mathematical City Information Transit Agent received the sealed theorem, Python reconstruction, numerical residuals, and correction distinction. The invariant cargo was preserved through routing.

</aside>

Canonical object

$$ \mathcal D=L_{\mathrm{FCC}}\otimes I_V+I_S\otimes K^2, \qquad \mathcal H=S\otimes V. $$

$$ \dim S=125,\qquad \dim V=125,\qquad E_{47}=\ker K,\qquad \dim E_{47}=47. $$

$$ \mathcal B_{47}:=S\otimes E_{47}, \qquad \Psi:=\ker\mathcal D. $$

Exact theorem state · E0

$$ \ker\mathcal D

\ker L_{\mathrm{FCC}}\otimes\ker K

\operatorname{span}\{\mathbf1_{\mathrm{FCC}}\}\otimes E_{47}. $$

$$ \dim\mathcal H=125\cdot125=15625, \qquad \dim\mathcal B_{47}=125\cdot47=5875, \qquad \dim\Psi=47. $$

$$ \Omega_{\mathrm{int}}

\frac{\dim E_{47}}{\dim V}

\frac{47}{125}

0.376. $$

$$ \Omega_{\mathrm{band}}

\frac{\dim\mathcal B_{47}}{\dim\mathcal H}

\frac{5875}{15625}

\frac{47}{125}

0.376. $$

$$ \Omega_{\ker}

\frac{\dim\Psi}{\dim\mathcal H}

\frac{47}{15625}

0.003008

0.3008\%. $$

<aside> ⚖️

CANONICAL RATIO DISTINCTION

$\Omega_{\mathrm{int}}=\Omega_{\mathrm{band}}=0.376$, while $\Omega_{\ker}=0.003008=\Omega_{\mathrm{int}}/125$.

The protected internal-band fraction is not the global asymptotic-kernel fraction.

</aside>

Valid and invalid cancellation

$$ \frac{125\cdot47}{125\cdot125}=\frac{47}{125} $$

$$ \frac{47}{125\cdot125}=\frac{47}{15625}\neq\frac{47}{125}. $$

Machine reconstruction · E1

Algebraic transit

$$ E_{47} \longmapsto S\otimes E_{47} \longmapsto \ker L_{\mathrm{FCC}}\otimes E_{47} $$